Review Quiz 2

We will be having a review in class every day until the next midterm exam on Wednesday of this week. For this quiz, you are given 10 minutes to decide which problems you think you can solve and which ones you want to see solved during our review session today.

Use the Lagrange Error Bound Formula for to find a reasonable bound for the error in approximating the quantity with a third-degree Taylor polynomial for the function about . Choose the best error estimate.

Solution: We use the formula

where on the interval (i.e., the interval between where our Taylor polynomial is centered and where we're approximating the value). Since has a maximum value of on the interval , we obtain

Use the Lagrange Error Bound Formula for to find a reasonable bound for the error in approximating the quantity with a third-degree Taylor polynomial for the function

about . Choose the best error estimate.

Solution: Here, we're approximating the given function using a third degree Taylor polynomial about zero, and we're plugging in since the function evaluated at that point is the value we're interested in. (And so the Taylor polynomial approximates when we plug in and find . How badly does the Taylor polynomial evaluated at approximate the true value of ? That's what we're looking for.

where on the interval . We find the fourth derivative to be

If you graph this function on , you find that it is always increasing. So, the maximum value is found (as in the previous example) at the rightmost point, which in this case is . So, we have . Thus,

For in the interval develop a formula (as a function of ) that yields the error bound for approximating the value of using the th Taylor series for centered about .

Solution: This is much like problem 1, but instead of plugging in a specific value of and a specific degree for the Taylor polynomial, the result will be a function of and . (This is basically a multivariable function. If you want to know how well the th degree Taylor polynomial of approximates at , this function will tell you a bound for the error.) We covered some of the details in class.

Prove that the Taylor series for centered at converges to for all real numbers .

Solution: The Taylor series is the “infinite degree” Taylor polynomial. So, we consider the limit of the error bounds for as . That is, we're looking at

Since the th derivatives of are always this becomes

We have noted several times in class that the limit on the right is zero. This shows that the difference between and the Taylor polynomials of degree as (the Taylor series) is zero. Thus, is actually equal to its Taylor series for all values of .

Prove that the Taylor series for centered at converges to for all real numbers .

Solution: The Taylor series is the “infinite degree” Taylor polynomial. So, we consider the limit of the error bounds for as . That is, we're looking at

Since all of the derivatives of satisfy , we know that . Thus, we have

We have noted several times in class that the limit on the right is zero. So, the proof is done. (The error always goes to zero, so the difference between the value of and the value of the Taylor series of is zero.)